詳解：拋物線方程式 \(y = {x^2} - x\) 可以寫成參數式 \(\left( {x\;,\;{x^2} - x} \right)\)

點 \((0,1)\) 到拋物線之距離函數為 \(f(x) = {(x - 0)^2} + {({x^2} - x - 1)^2} = {x^2} + {({x^2} - x - 1)^2}\)

\(f(x) = {(x - 0)^2} + {({x^2} - x - 1)^2} = {x^2} + {({x^2} - x - 1)^2}\)

\(f'(x) = 2x + 2({x^2} - x - 1)(2x - 1)\) 展開後分解因式

\(f'(x) = 2x + 2({x^2} - x - 1)(2x - 1) = 4{x^3} - 6{x^2} + 2 = (x - 1)(4{x^2} - 2x - 2)\)

\(f'(x) = {(x - 1)^2}(4x + 2)\)

臨界值 \(x = 1\) ， \(x =  - \frac{1}{2}\)

\(f(1) = {1^2} + {(1 - 1 - 1)^2} = 2\)

\(f( - \frac{1}{2}) = \frac{1}{4} + {(\frac{1}{4} + \frac{1}{2} - 1)^2} = \frac{1}{4} + \frac{1}{{16}} = \frac{5}{{16}}\)

就是當 \(x =  - \frac{1}{2}\) ，出現極小值 \(\sqrt {\frac{5}{{16}}} {\rm{ = }}\frac{{\sqrt {\rm{5}} }}{{\rm{4}}}\)

故選(D)